Computer simulations are a widely used tool in computational science and engineering to, e.g., analyze the behavior of components or materials, to enhance the development of manufacturing processes with fast and accurate a-priori forecasts, or even to control those processes during ongoing operation. The underlying models need to be capable of covering all relevant effects while fulfilling the accuracy required for the analysis. These so-called high-fidelity models are often based on methods like the Finite Element Method (FEM). With the help of these tools, we wish to make reliable assertions and predictions for one or more quantity of interests (QoI), also in the presence of uncertainty, e.g., in process conditions, material properties or similar influencing factors. Thus, methods from the field of Uncertainty Quantification (UQ) can enhance the quality of processes and products by augmenting the results for the QoI with quantified probability measures, eventually leading to an increased reliability and an additional support in decision-making. In this context, employing the original, high-fidelity model may easily exceed available resources or required feedback times. Here, the usage of surrogate models, which are computationally cheaper, can provide a remedy. To that end, we investigate two strategies. First, we consider sampling-based UQ methods that usually require a great number of evaluations of the underlying model. Therefore, instead of relying on the high-fidelity model, we follow the path of Model Order Reduction (MOR). Based on the original FEM model, a reduced model with decreased computational complexity is constructed, while keeping its accuracy in the desired range. In particular, we explore the benefits of intrusive MOR techniques. The approach that will be presented makes use of Proper Orthogonal Decomposition (POD), where a subsequent Galerkin projection of the operators onto the constructed subspaces is performed. As an alternative, one also may take a probabilistic view point from the outset in constructing the surrogate model. In this spirit, we investigate the advantages of Gaussian Process Regression (GPR) as a meta model. Following a Bayesian approach, we compute a predictive distribution for the QoI, given certain training data. The latter can be based on samples produced by the high-fidelity FEM model but also on measurements. Finally, the integration of the resulting surrogate models into an UQ setting is demonstrated for applications coming from polymer processing or material analysis.